The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues

Size: px

Start display at page:

Download "The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues"

Elfrieda Johns
6 years ago
Views:

1 enedikt et al. ounday Value Poblems 2011, 2011:27 RESEARCH Open Access The fist nontivial cuve in the fučĺk spectum of the diichlet laplacian on the ball consists of nonadial eigenvalues Jiřĺ enedikt 1*, Pavel Dábek 2 Pet Gig 1 * Coespondence: benedikt@kma. zcu.cz 1 Depatment of Mathematics, Faculty of Applied Sciences, Univesity of West ohemia, Univezitnĺ 22, Plzeň, Czech Republic Full list of autho infomation is available at the end of the aticle Abstact It is well-known that the second eigenvalue l 2 of the Diichlet Laplacian on the ball is not adial. Recently, atsch, Weth Willem poved that the same conclusion holds tue fo the so-called nontivial sign changing Fučík eigenvalues on the fist cuve of the Fučík spectum which ae close to the point l 2, l 2. We show that the same conclusion is tue in dimensions 2 3 without the last estiction. Keywods: Fučík spectum, The fist cuve of the Fučík spectum, Radial nonadial eigenfunctions 1. Intoduction Let Ω R N be a bounded domain, N 2. The Fučík spectum of -Δ on W 1,2 0 is defined as a set Σ of those l +, l - Î R 2 such that the Diichlet poblem u = λ+ u + λ u in, 1 u =0 on has a nontivial solution u W 1,2 0. In paticula, if l 1 < l 2 <... ae the eigenvalues of the Diichlet Laplacian on Ω counted with multiplicity, then clealy Σ contains each pai l k, l k, k Î N, the two lines l 1 } R R l 1 }. Following [1, p. 15], we call the elements of Σ \l 1 } R R l 1 } nontivial Fučík eigenvalues. Itwas poved in [2] that thee exists a fist cuve C of nontivial Fučík eigenvalues in the sense that, defining h: l 1, R by ηλ def inf μ>λ 1 :λ, μ is a nontivial Fučík eigenvalue }, we have that l 1 < hl < fo evey l >l 1, the cuve C def λ, ηλ : λ λ 1, } consists of nontivial Fučík eigenvalues. Moeove, it was poved in [2] that C is a continuous stictly deceasing cuve which contains the point l 2, l 2 which is symmetic with espect to the diagonal. Itwasconjectuedin[1,p.16],thatifΩ is a adially symmetic bounded domain, then evey eigenfunction u of 1 coesponding to some λ +, λ Cis not adial. The 2011 enedikt et al; licensee Spinge. This is an Open Access aticle distibuted unde the tems of the Ceative Commons Attibution License which pemits unesticted use, distibution, epoduction in any medium, povided the oiginal wok is popely cited.

2 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 2 of 9 authos of [1, p. 16] actually poved that the conjectue is tue if λ +, λ Cbut sufficiently close to the diagonal. The oiginal pupose of this pape was to pove that the above conjectue holds tue fo all λ +, λ C povided Ω is a ball in R N with N =2N = 3. Without loss of geneality, we pove it fo the unit ball cented at the oigin. Cf. Theoem 6 below. Duing the eview of this pape, one of the eviewes dew the authos attention to the pape [3], whee the same esult is poved fo geneal N 2see[3,Theoem 3.2]. The poof in [3] uses the Mose index theoy coves also poblems with weights on moe geneal domains than balls. On the othe h, ou poof is moe elementay geometically instuctive. Fom this point of view, ou esult epesents a constuctive altenative to the athe abstact appoach pesented in [3]. This is the main authos contibution. 2. Vaiational chaacteization of C Let us fix s Î R let us daw in the l +, l - plane a line paallel to the diagonal passing though the point s, 0, see Figue 1. We show that the point of intesection of this line C coesponds to the citical value of some constained functional cf. [4, p. 214]. To this end we define the functional J s u def u 2 s u + 2. Then J s u is a C 1 -functional on W 1,2 0 we look fo the citical points of the estiction J s of J s to S def u W 1,2 0 } :Iudef u 2 =1. y the Lagange multiplies ule, u S is a citical point of J s if only if thee exists t Î R such that λ C diagonal paallel λ 1 0 λ 1 s λ + Figue 1 The fist two Fučík cuves.

3 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 3 of 9 J s u =ti u, i.e., u v s u + v = t uv 2 fo all v W 1,2 0. This means that u =s + tu + tu in, u =0 on holds in the weak sense. In paticula, l +, l - =s + t, t Î Σ. Takingv = u in 2, one can see that the Lagange multiplie t is equal to the coesponding citical value of J s. Fom now on we assume s 0, which is no estiction since Σ is clealy symmetic with espect to the diagonal. The fist eigenvalue l 1 of -Δ on W 1,2 0 is defined as λ 1 = λ 1 def min u 2 : u W 1,2 0 u 2 =1. 3 It is well known that l 1 >0, simple admits an eigenfunction ϕ 1 W 1,2 0 C1 with 1 satisfying 1 x >0 fo x Î Ω. Let Ɣ def γ C[ 1, 1], S :γ 1 = ϕ 1 γ 1 = ϕ 1 } cs def inf max γ Ɣ u γ J s u. 4 We keep the same notation g fo the image of a function g = g t. It follows fom [4, Pops. 2.2, 2.3 Thms. 2.10, 3.1] that the fist thee citical levels of J s ae classified as follows. i 1 is a stict global minimum of J s with J s ϕ 1 = λ 1 s. The coesponding point in Σ is l 1, l 1 - s, which lies on the vetical line though l 1, l 1. ii - 1 is a stict local minimum of J s,j s ϕ 1 = λ 1. The coesponding point in Σ is l 1 + s, l 1, which lies on the hoizontal line though l 1, l 1. iii Fo each s 0, the point s + cs, cs, whee cs > l 1 is defined by the minimax fomula 4, belongs to Σ. Moeove, the point s + cs, cs is the fist nontivial point of Σ on the paallel to the diagonal though s, 0. Next we summaize some popeties of the dependence of the pincipal fist eigenvalue l 1 Ω on the domain Ω. The following poposition follows immediately fom the vaiational chaacteization of l 1 given by 3 the popeties of the coesponding eigenfunction 1. Poposition 1. l 1 Ω 2 < l 1 Ω 1 wheneve Ω i,i=1,2,ae bounded domains satisfying Ω 1 Ω 2 measω 1 <measω 2. Let us denote by V d, d Î 0, 1, the ball canopy of the height 2d by d the maximal inscibed ball in V d see Figue 2. It follows fom Poposition 1 that fo d Î 0, 1, we have

4 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 4 of 9 y d 2d V d V 1 d d x 1 1 d Figue 2 The ball decomposition λ 1 V d <λ 1 d, λ 1 V 1 d <λ 1 1 d. 5 Moeove, fom the vaiational chaacteization 3, the following popeties of the function d λ 1 V d 6 follow immediately. Poposition 2. The function 6 is continuous stictly deceasing on 0, 1, it maps 0, 1 onto l 1, lim λ 1 V d d 0+ =, lim λ 1V d =λ 1. d 1 In paticula, it follows fom Poposition 2 that, given s 0, thee exists a unique d s 0, 1 2 ] such that λ 1 V ds =s + λ 1 V 1 ds. 7 Let u ds u 1 ds be positive pinciple eigenvalues associated with λ 1 V ds λ 1 V 1 ds, espectively. We extend both functions on the entie by setting u ds 0 on u 1 ds 0, u 1 ds 0 on V ds then nomalize them by u ds, u 1 ds S. Ouaimisto constuct a special cuve g Î Γ on which the values of J s stay below λ 1 V ds. Actually, the cuve g connects 1 with - 1 passes though u ds u 1 ds. Fo this pupose we set g = g 1 g 2 g 3, whee } def γ 1 u =τϕ τu2 d s 2 : τ [0, 1], def γ 2 u = αu ds βu 1 ds : α 0, β 0, α 2 + β 2 =1 }, } def γ 3 u = τϕ τu2 1 d s 2 : τ [0, 1]. Changing suitably the paametization of g i, i =1,2,3weskipthedetailsfothe bevity, g can be viewed as a gaph of a continuous function, mapping [-1, 1] into S. We pove Poposition 3. J s u λ 1 V 1 ds fo all u Î g.

5 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 5 of 9 Fo the poof we need so-called ay-stict convexity of the functional J v def v defined on } def V + v : 0, :v 1 2 W 1,2 0 C. We say that J : V + R is ay-stictly convex if fo all τ Î 0, 1 v 1, v 2 Î V + we have J 1 τv 1 + τv 2 1 τj v 1 + τj v 2 whee the equality holds if only if v 1 v 2 ae colinea. Lemma 4 see [5, p. 132]. The functional J defined by 8 is ay-stictly convex. Poof of Poposition The values on g 1. Fo u Î g 1 we have J s u =J u 2 s u 2 = τϕ τu2 2 d s s τϕ τu 2 d s τ ϕ τ u ds 2 s τ ϕ τ τ u ds 2 +1 τ u ds 2 s u ds 2 s = λ 1 V ds s = s + λ 1 V 1 ds s = λ 1 V 1 ds V ds by Lemma 4 with Ω :=, The values on g 2.Letu Î g 2, then thee exist a 0, b 0, a 2 + b 2 =1such that u = αu ds βu 1 ds.sincethesuppotsofu ds u 1 ds ae mutually disjoint, we have J s u =α 2Vds u ds 2 + β 2V1 ds u 1 ds 2 α 2 svds u 2 d s u 2 d s = α 2 λ 1 V ds +β 2 λ 1 V 1 ds α 2 s = α 2 s +α 2 + β 2 λ 1 V 1 ds α 2 s = λ 1 V 1 ds by The values on g 3. Fo u Î g 3 we have similaly as in the fist case J s u = τϕ τu2 2 1 d s u 1 ds 2 = λ 1 V 1 ds. V 1 ds Fom Poposition 3, 4 5 we immediately get Poposition 5. Given s 0, we have cs λ 1 V 1 ds <λ 1 1 ds. 9

6 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 6 of 9 3. Radial eigenfunctions Radial Fučík spectum has been studied in [6]. Let x be the Euclidean nom of x Î R N u = u x be a adial solution of the poblem u = λ+ u + λ u in, 10 u =0 on. Set = x witev =u x. It follows fom the egulaity theoy that 10 is equivalent to the singula poblem v + λ + v + λ v = 0 in 0, 1, v 11 0 = 0, v1 = 0. The authos of [6] povide a detailed chaacteization of the Fučík spectum of 11 by means of the analysis of the linea equation associated to 11: v + N 1 v + λv =0 in0, The function v is a solution of 12 if only if ˆv = 2 N 1 v is a solution of ˆv + λ + N 13 N 4 2 ˆv =0 in0,. 13 Note that the functions v ˆv have the same zeos. Let us investigate the adial Fučík eigenvalues which lie on the line paallel to the diagonal which passes though the point s, 0 in the l +, l - -plane. The fist two intesections coincide with the points l 1, l 1 - s l 1 + s, l 1. This fact follows fom the adial symmety of the pincipal eigenfunction of the Diichlet Laplacian on the ball. A nomalized adial eigenfunction associated with the next intesection has exactly two nodal domains it is eithe positive o else negative at the oigin. Let us denote the fome eigenfunction by u 1 the latte one by u 2, espectively. Let l 1 + s, l 1 l 2 + s, l 2 befučík eigenvalues associated with u 1 u 2, espectively. The popety iii on page 5 implies that cs l i, i =1,2. The main esult of this pape states that the above inequalities ae stict it is fomulated as follows. Theoem 6. Let N =2o N =3 s Î R be abitay. Then cs <λ i, i =1,2. In paticula, nontivial Fučík eigenvalues on the fist cuve of the Fučík spectum ae not adial. Poof. Letu i x =v i, i =1,2, = x. Then thee exists d 1 Î 0, 1 such that v 1 is a solution of v +s + λ 1 v =0 v > 0 in 0,d 1, v 0 = vd 1 =0

7 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 7 of 9 v + λ 1 v =0 v < 0 in d 1,1, vd 1 =v1 = 0. Afte the substitution ˆv 1 = 1 2 N 1 v 1, ˆv 1 is a solution of ˆv + s + λ 1 + ˆv0 = ˆvd 1 =0 ˆv + λ 1 + ˆvd 1 =ˆv1 = 0. N 13 N 4 2 ˆv =0 ˆv > 0 in 0,d 1, N 13 N 4 2 ˆv =0 ˆv < 0 in d 1,1, Let u 1 = u 1 x u 2 = u 2 x be the pincipal positive eigenfunctions associated with λ 1 ds λ 1 1 ds, espectively. oth u i, i = 1, 2, ae adially symmetic with espect to the cente of the coesponding ball. Due to the invaiance of the Laplace opeato with espect to tanslations we may assume that both ds 1 ds ae cented at the oigin. We then set u i x =w i, i =1,2, = x. The functions w i, i = 1, 2, solve w 1 + N 1 w 1 + λ 1 ds w 1 =0 w 1 > 0 in 0,d s, w 10 = w 1 d s =0 w 2 + N 1 w 2 + λ 1 1 ds w 2 =0 w 2 > 0 in 0,1 d s, w 20 = w 2 1 d s =0. Afte the substitution ŵi = 1 2 N 1 w i, i = 1, 2, we have ŵ 1 + λ 1 ds + ŵ 1 0 = ŵ 1 d s =0 N 13 N 4 2 ŵ 1 =0 ŵ 1 > 0 in 0,d s, 16 ŵ 2 + λ 1 1 ds + ŵ 2 0 = ŵ 2 1 d s =0. N 13 N 4 2 ŵ 2 =0 ŵ 2 > 0 in 0,1 d s, The substitution ṽ = ˆv + d 1 tansfoms 15 to ṽ + λ 1 N 13 N d 1 2 ṽ =0 ṽ > 0 in 0,1 d 1, ṽ0 = ṽ1 d 1 =0. 17 Let us assume that λ 1 λ 1 V 1 ds <λ 1 1 ds that d 1 > d s. Choose δ = d1 d s 2 set w 2 = ŵ 2 + δ. Then w 2 solves

8 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 8 of 9 w 2 + λ 1 1 ds + w 2 δ = w 2 1 d s δ =0. N 13 N 4 + δ 2 w 2 =0 w 2 > 0 in δ,1 d s δ, 18 It follows that 18 is a Stum majoant fo 17 on the inteval I =[ δ 2,1 d s δ 2 ] w 2 > 0 on J.Sinceṽ0 = ṽ1 d 1 = 0 0 I, 1 d 1 I, wehaveacontadiction with the Stum Sepaation Theoem see [7, Co. 3.1, p. 335]. Hence d 1 d s. Simila application of the Stum Sepaation Theoem to now yields λ 1 ds s + λ Since we also have λ 1 ds >λ 1 V ds, it follows fom 7 19 that s + λ 1 V 1 ds =λ 1 V ds <λ 1 ds s + λ 1 s + λ 1 V 1 ds, a contadiction which poves that λ 1 >λ 1 V 1 ds. Similaly as above, thee exists d 2 Î 0, 1 such that v 2 is a solution of v + λ 2 v =0 v < 0 in 0,d 2, v 0 = vd 2 =0 v +s + λ 2 v =0 v > 0 in d 2,1, vd 2 =v1 = 0. Afte the substitution ˆv 2 = 1 2 N 1 v 2, ˆv 2 is a solution of ˆv + λ 2 + ˆv0 = ˆvd 2 =0 ˆv + s + λ 2 + ˆvd 2 =ˆv1 = 0. N 13 N 4 2 ˆv =0 ˆv < 0 in 0,d 2, N 13 N 4 2 ˆv =0 ˆv > 0 in d 2,1, Assume that λ 2 λ 1 V 1 ds <λ 1 1 ds that 1- d s >d 2. Simila aguments based on the Stum Compaison Theoem yield fist that 1- d s d 2 i.e., 1 - d 2 d s, then 16, 21 that λ 1 ds s + λ 2. As above we obtain s + λ 1 V 1 ds =λ 1 V ds <λ 1 ds s + λ 2 s + λ 1 V 1 ds, a contadiction which poves that λ 2 >λ 1 V 1 ds. The assetion now follows fom Poposition 5. Remak 7. Caeful investigation of the above poof indicates that N -13-N 0 is needed to make the compaison aguments wok. The poof is simple fo N =3

9 enedikt et al. ounday Value Poblems 2011, 2011:27 Page 9 of 9 when the tansfomed equations fo ˆv ŵ ae autonomous. The application of the Stum Compaison Theoem is then moe staightfowad. Acknowledgments Jiří enedikt Pet Gig wee suppoted by the Poject KONTAKT, ME 10093, Pavel Dábek was suppoted by the Poject KONTAKT, ME Autho details 1 Depatment of Mathematics, Faculty of Applied Sciences, Univesity of West ohemia, Univezitnĺ 22, Plzeň, Czech Republic 2 Depatment of Mathematics N.T.I.S., Faculty of Applied Sciences, Univesity of West ohemia, Univezitnĺ 22, Plzeň, Czech Republic Authos contibution All authos contibuted to each pat of this wok equally. Competing inteests The authos declae that they have no competing inteests. Received: 3 May 2011 Accepted: 4 Octobe 2011 Published: 4 Octobe 2011 Refeences 1. atsch, T, Weth, T, Willem, M: Patial symmety of least enegy nodal solutions to some vaiational poblems. J. D Analyse Mathématique. 96, de Figueiedo, D, Gossez, J-P: On the fist cuve of the Fučík spectum of an elliptic opeato. Diffe. Integal Equ. 7, atsch, T, Degiovanni, M: Nodal solutions of nonlinea elliptic Diichlet poblems on adial domains. Rend. Licei Mat. Appl. 17, Cuesta, M, de Figueiedo, D, Gossez, J-P: The beginning of the Fučík spectum fo the p-laplacian. J. Diffe. Equ. 159, doi: /jdeq Takáč, P: Degeneate elliptic equations in odeed anach spaces applications. In: Dábek P, Kejčí P, Takáč P eds. Nonlinea Diffeential Equations. Chapman Hall/CRC Res. Notes Math, vol. 404, pp CRC Pess LLC, oca Raton Aias, M, Campos, J: Radial Fučik spectum of the Laplace opeato. J. Math. Anal. Appl. 190, doi: /jmaa Hatman, P: Odinay Diffeential Equations. Wiley, New Yok 1964 doi: / Cite this aticle as: enedikt et al.: The fist nontivial cuve in the fučĺk spectum of the diichlet laplacian on the ball consists of nonadial eigenvalues. ounday Value Poblems :27. Submit you manuscipt to a jounal benefit fom: 7 Convenient online submission 7 Rigoous pee eview 7 Immediate publication on acceptance 7 Open access: aticles feely available online 7 High visibility within the field 7 Retaining the copyight to you aticle Submit you next manuscipt at 7 spingeopen.com

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming